Four athletes run a race, starting from the same point and all of them

Question

Four athletes run a race, starting from the same point and all of them running clockwise. If the ratio of the speeds of the athletes are 1 : 2 : 3 : 4, at how many distinct points on the circular track could any two athletes meet (overtake)?

A. 4
B. 5
C. 6
D. 8
E. 10

A. 4 
B. 5 
C. 6 
D. 8 
E. 10

rajatchopra1994 · Accepted Answer

Explanation:

There would be 4C2 = 6 different combinations taking two at a time. Let’s take them one by one.

1. A and B: The speeds are in the simplest form and number of distinct points they meet = 2-1 = 1.
2. A and C: The speeds are in the simplest form and number of distinct points they meet = 3-1 = 2.
3. A and D: The speeds are in the simplest form and number of distinct points they meet = 4-1 = 3.
4. B and C: The speeds are in the simplest form and number of distinct points they meet = 3-2 = 1.
5. B and D: The speeds are NOT in the simplest form. To reduce the same into the simplest form, we will take their simplest ratio. i.e. 2 : 4 = 1:2. Therefore, number of distinct points they meet = 2-1 = 1.
6. C and D: The speeds are in the simplest form and number of distinct points they meet = 4-3 = 1.
We know that the last meeting point will always be at the starting point. So when any two of them meet only for one time that will be at the starting point only.
When they meet for the two times, one will be at the starting point and other at a point diametrically opposite to the starting point. Similarly, when they meet three times, one will be at the starting point and rest two points will be at 120 degrees with the starting point.
Hence in total, they will meet at 4 distinct points.

IMO-A

srishti246 · Answer

Could someone please provide another explanation to this question? I am unable to understand the solution provided.

ishita27 · Answer

Can someone give another explanation for this question? 
 Bunuel

Fdambro294 · Answer

this question is based on an Obscure Rule that I do not believe is relevant to the GMAT.

Concept: If the Ratio of the Speeds of 2 Runners --- Runner A and Runner B---- running in the SAME Direction around a Circular Track = p : q ----- where p/q Ratio is in its Lowest Canceled Form

then A and B will meet at (p - q) Distinct Points on the Circular Track AND the (p - q) Meeting Points around the Circular Track will evenly Divide the Circle into Equal Arc Lengths.

Further Rule: the LAST DISTINCT Meeting Point will always be the Starting Point.

Let the following Ratio be:

Speed of Runner A : Speed of Runner B = 1 : 2

Runner A and Runner B will only meet at (2 - 1) = 1 Distinct Meeting Point (which will be at the Start)

To check this, Let Speed of A = 5 m.p.s. and Let Speed of B = 10 m.p.s.

and the Circular Track = 100 miles

Time until 1st Meet = 100 / 10 - 5 = 20 Seconds.

In 20 Seconds, A will travel: 5 * 20 = 100 miles, which is 1 Lap around the track right back to the Starting Point

In 20 Seconds, B will travel: 10 * 20 = 200 miles, which is 2 Laps around the track right back to the Starting Point.

This is the only DISTINCT Meeting Place that Runners A and B will ever Meet at on the Circular Track right at the Starting Point. Every other meeting point from that instant on will take place again right at the Starting Point.

Now performing the Same Calculation for all the Runners:

Speed A : Speed B : Speed C : Speed D = 1 : 2 : 3 : 4

B and A will meet at (2 - 1) = 1 Distinct Meeting Point (the Start)

C and A will meet at (3 - 1) = 2 Distinct Meeting Points - 1/2 way around the Circle and at the Start

Since we already counted the Start, we now have 2.

D and A will meet at (4 - 1) = 3 Distinct Meeting Points - 1/3rd way around the Circle, 2/3rd way around the Circle, and back at the Start

These is 2 More Distinct Meeting Points. We now have 4

C and B will meet at (3 - 2) = 1 Distinct Meeting Point - which is the Starting Point

we already counted this Meeting Point

D and B will meet at (2 -1) = 1 Distinct Meeting Point - which is the Starting Point ***(remember, the Ratio p:q has to be in its lowest form - so reducing 4:2 --- to 2:1)

we already counted this meeting Point

D and C will meet at (4 - 3) = 1 Distinct Meeting Point - which is the Starting Point

we already counted this meeting point.

Answer- there are 4 Distinct Meeting Points on the Circular Track at which any 2 of the 4 athletes could meet each other.

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Four athletes run a race, starting from the same point and all of them

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